Question


Constructed Response Question
King Middle School has three grade levels. The associated student body, a club on campus, has decided that they would like to host an after-school dance to raise money. The table shows the number of students in each of these grade levels.



If the associated student body decides to host this dance, their goal is to have a total of 261 students attend. If the seventh-grade class and the eighth-grade class exactly meet their goals for attendance, what percent of sixth-grade students would need to attend for the club to meet its goals? Explain or show your reasoning. You may use a combination of diagrams, drawings, expressions/equations, and words.

If the dance occurs, 261 students will need to attend to reach the goal of raising at least $1174.50. Each student will raise $4.50 for attending. Together, the seventh-grade and eighth-grade students expect to raise $787.50.

The president of the associated student body, Amalia, wrote and solved the following inequality to determine x, the number of sixth-grade students that will need to attend the dance.

4.50x ≥ 787.50
x ≥ 175

Do you think that Amalia correctly determined the number of sixth-grade students who need to attend?

If your answer is yes, provide a justification. If your answer is no, provide a justification by writing and solving the correct inequality.











Answers

Answered by GPT-4o mini
To determine what percent of sixth-grade students need to attend the dance for the club to meet its attendance goal of 261 students, we first need to understand how many students are anticipated from the seventh and eighth grades.

Let's break this down with the information provided:

1. **Attendance Goal:** 261 students
2. **Amount raised per student:** $4.50
3. **Amount raised by seventh and eighth graders combined:** $787.50

**Step 1: Calculate how many seventh and eighth-grade students are expected to attend.**

To find out how many students from the seventh and eighth grades are attending, we can calculate how many total funds would be raised by those grades and how many students it represents:

Each student raises $4.50, so the total number of students from seventh and eighth grades, combined, can be calculated as follows:

\[
\frac{787.50}{4.50} = 175
\]

This means that the seventh and eighth grades are expected to have a total attendance of 175 students.

**Step 2: Calculate how many sixth-grade students need to attend.**

To figure out how many sixth-grade students are required, we'll subtract the attendance from the seventh and eighth grades from the total attendance goal:

\[
\text{Sixth-grade attendance needed} = \text{Total attendance goal} - \text{Seventh and eighth-grade attendance}
\]

\[
\text{Sixth-grade attendance needed} = 261 - 175 = 86
\]

**Step 3: Determine the percentage of sixth-grade students that need to attend.**

Let \( S \) represent the total number of sixth-grade students. The percent of sixth-grade students that need to attend can be represented as:

\[
\text{Percent of sixth-graders attending} = \left( \frac{\text{Number of sixth-graders attending}}{S} \right) \times 100
\]

Since we found that 86 sixth graders need to attend:

1. If \( S = \text{Total sixth-grade students} \), we can now find the percentage of sixth graders attending based on the number \( S \).

The specific percent cannot be calculated accurately without the exact number of sixth-grade students. If, for example, the sixth-grade class had 100 students, then:

\[
\text{Percent} = \left( \frac{86}{100} \right) \times 100 = 86\%
\]

But if the sixth grade had 150 students:

\[
\text{Percent} = \left( \frac{86}{150} \right) \times 100 \approx 57.33\%
\]

So, to determine the exact percent, we need the actual number of sixth-grade students.

**Step 4: Determine if Amalia’s calculation is correct.**

Amalia's inequality reads:

\[
4.50x \geq 787.50
\]

Her calculation suggests the inequality for sixth-grade students attending is based solely on the funds raised. However, what she needs to determine is how many sixth graders should attend to meet the total attendance goal of 261.

The correct inequality should consider the total attendance:

\[
x + 175 = 261 \Rightarrow x = 261 - 175 \Rightarrow x = 86,
\]

Which means 86 sixth-grade students are required to achieve the goal of 261 total attendees.

Thus, Amalia's approach incorrectly tied the number of sixth graders to funds when it should have strictly resolved the attendance goal. Therefore, she did not determine the correct number of sixth graders attending the dance.
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